18.099: Problem Set 5

Due: Tuesday, October 22.

The purpose of this problem set is to write an exposé about the p-adic integers. You are free to organize the material as you see fit.

Let p be a prime number. We define a p-adic integer a to be a sequence a = (a_{0}, a_{1}, a_{2}, …) of integers 0 ≤ a_{i} < p. It is common to display the sequence a as an infinite “sum”

a = a_{0} + a_{1}p + a_{2}p^{2} + a_{3}p^{3} + · · · .

Every non-negative integer n can be written uniquely as a sum

n = n_{0} + n_{1}p + n_{2}p^{2} + n_{3}p^{3} + ⋯,

with 0 ≤ n_{i} < p and with all but finitely many n_{i} equal to zero. This way we obtain an injective map

ι: ℕ_{0} ↪ ℤ_{p}

from the set of non-negative integers to the set of p-adic integers. We define sum and product of p-adic integers by the usual formulas such that ι(m + n) = ι(m) + ι(n) and ι(m · n) = ι(m) · ι(n). Show that

−1 = (p − 1) + (p − 1) · p + (p − 1) · p^{2} + (p − 1) · p^{3} + ⋯

such that we can extend the map ι to an injective map

ι: ℤ ↪ ℤ_{p}

by sending a negative integer −n to −1 · ι(n). In the following, if n is an integer, we will sometimes abuse notation and also write n for the p-adic integer ι(n).

Let a be a p-adic integer. We define 0 ≤ v_{p}(a) ≤ ∞ to be the smallest i for which a_{i} ≠ 0. Show that v_{p}(a) is equal to the number of times that p = ι(p) divides a. We define the p-adic metric on the set ℤ_{p} by

d_{p}(a, b) = \begin{cases}p^{−v_{p}(a−b)} & if a ≠ b, \\ 0 & if a = b. \end{cases}

Show that d_{p} satisfies the stronger triangle inequality

d_{p}(a, c) ≤ max{d_{p}(a, b), d_{p}(b, c)},

and conclude that d_{p} is a metric. A metric that satisfies the stronger triangle inequality is called an ultra-metric. Show that all triangles in an ultra-metric space are isosceles. Show that the metric space (ℤ_{p}, d_{p}) is complete.

Let B_{r}(a) = {b ∈ ℤ_{p} | d_{p}(a, b) < r} be the open ball of radius r > 0 around a. Prove the following statements.

(i) If b ∈ B_{r}(a), then B_{r}(a) = B_{r}(b).

(ii) The subset B_{r}(a) ⊂ ℤ_{p} is both open and closed.

(iii) The intersection B_{r}(a) ∩ B_{s}(b) is non-empty if and only if one of the two balls contains the other.

A metric space (X, d) is called totally bounded if for every ϵ > 0, there exists a finite cover of X be open balls of radius ϵ. Show that (ℤ_{p}, d_{p}) is totally bounded. Show that a metric space (X, d) is compact if and only if it is complete and totally bounded. (The “if” part is the more difficult. Show that every sequence {x_{n}} in X has a subsequence that is Cauchy.) Conclude that (ℤ_{p}, d_{p}) is compact.